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I guess the set $q-C$ is fairly simple, so a resource-bounded random real (like a polynomial time random one) should suffice. Perhaps one of those is even sort of explicitly definable, like a "polynomial time $\Omega$"...
Here is the usual recursion theoretical comment: I gather that if $A_{\langle\sigma,n\rangle,\tau}$ says $\sigma$ is a prefix of $\tau$ and $\tau$ belongs to or strongly avoids a certain dense set $D_n$, then François' last equation says there is an $f$ that given $\sigma$ and $n$ finds such a $\tau$; if the sequence $D_n$ is the collection of $\Sigma^0_1$ sets then $f$ computes a 1-generic $G$. Interesting...
Oh okay, so if we think in terms of propositional logic with variables $p_1,p_2,\ldots$ then an atom would correspond to a complete truth assignment on all the variables, but (1) in the Cohen algebra a complete truth assignment gets identified with 0, and (2) if $\mathbb P(p_n=\text{True})=1/2$ then any particular truth assignment has probability 0, i.e., is Lebesgue negligible, and that's the connection with @Simon Henry's answer. Thanks.
